Unless otherwise noted, the content of this course material is
licensed under a Creative Commons Attribution 3.0 License.
http://creativecommons.org/licenses/by/3.0/
© 2009, Peter Von Buelow
You assume all responsibility for use and potential liability associated with any use of the material. Material contains copyrighted content, used in
accordance with U.S. law. Copyright holders of content included in this material should contact open.michigan@umich.edu with any questions,
corrections, or clarifications regarding the use of content. The Regents of the University of Michigan do not license the use of third party content
posted to this site unless such a license is specifically granted in connection with particular content. Users of content are responsible for their
compliance with applicable law. Mention of specific products in this material solely represents the opinion of the speaker and does not represent an
endorsement by the University of Michigan. For more information about how to cite these materials visit
https://open.umich.edu/education/about/terms-of-use.
Any medical information in this material is intended to inform and educate and is not a tool for self-diagnosis or a replacement for medical
evaluation, advice, diagnosis or treatment by a healthcare professional. You should speak to your physician or make an appointment to be seen if you
have questions or concerns about this information or your medical condition. Viewer discretion is advised: Material may contain medical images that
may be disturbing to some viewers.
Architecture 324
Structures II
Reinforced Concrete - WSD
• Material Properties
• Stress in Beams
• Transformed Sections
• Analysis by WSD
• Design by WSD
University of Michigan, TCAUP
Structures II
Slide 2/23
Constituents of Concrete
•
•
•
•
Sand
Aggregate
Cement
Water
Fine aggregate
(Sand)
≤ 1/4”
~ 3/8” aggregate
limestone aggregate
~ 1.5”
Photos: CC:BY-SA Emadrazo (wikipedia)
http://creativecommons.org/licenses/by-sa/3.0/
University of Michigan, TCAUP
Structures II
Slide 3/23
Constituents of Concrete
Cement Types
•
•
•
Sand
Aggregate
Water
•
Cement
• Limestone
• Cement rock
• Clay
• Iron ore
• + (after firing and grinding)
• gypsum
• Type 1
normal portland cement. Type 1 is a
general use cement.
• Type 2
is used for structures in water or soil
containing moderate amounts of sulfate,
or when heat build-up is a concern.
• Type 3
high early strength. Used when high
strength are desired at very early periods.
• Type 4
low heat portland cement. Used where the
amount and rate of heat generation must
be kept to a minimum.
• Type 5
Sulfate resistant portland cement. Used
where water or soil is high in alkali.
• Types IA, IIA and IIIA are cements used to
make air-entrained concrete.
University of Michigan, TCAUP
Structures II
Slide 4/23
Workability
• Measured by the inches of “slump” of
a molded cone of fresh mix.
– range 1” to 4” with vibration
– 2” to 6” without vibration
• Water/Cement Ratio
– range 0.4 to 0.7
– for strength: higher is weaker
– for workability: higher is better
• Cement Content
–
–
–
–
LBS per cubic yard
range 400-800
dependent on aggregate
increases cost
Photos: CC:BY-SA Tano (wikipedia)
http://creativecommons.org/licenses/by-sa/3.0/
University of Michigan, TCAUP
Structures II
Slide 5/23
Reinforcing
• Grade = Yield strength
•
•
gr. 40 is 40 ksi
gr. 60 is 60 ksi
• Size in 1/8 inch increments
•
•
#4 is ½ inch dia.
#6 is ¾ inch dia.
• Deformation Patterns
•
add to bond with concrete
• Spacing
•
between bars
Bar diameter
1”
5/4 x max agg.
•
between layers
1”
•
coverage
3” against soil
1.5”-2” exterior
3/4” interior
University of Michigan, TCAUP
Reinforcement of Weidatalbrücke
CC:BY-SA Störfix (wikipedia)
http://creativecommons.org/licenses/by-sa/3.0/
Structures II
Slide 6/23
Curing
Strength increases with age. The
“design” strength is 28 days.
Source: Portland Cement Association
University of Michigan, TCAUP
Structures II
Slide 7/23
Strength Measurement
• Compressive strength
– 12”x6” cylinder
– 28 day moist cure
– Ultimate (failure) strength
f c'
•Tensile strength
–
–
–
–
–
'
12”x6” cylinder
f
t
28 day moist cure
Ultimate (failure) strength
Split cylinder test
Ca. 10% to 20% of f’c
Photos: Source: Xb-70 (wikipedia)
University of Michigan, TCAUP
Structures II
Slide 8/23
Young’s Modulus
• Depends on density and strength
Ec  wc1.5 33 f c'
• For normal (144 PCF) concrete
Ec  57000 f c'
• Examples
f’c
3000 psi
4000 psi
5000 psi
E
3,140,000 psi
3,620,000 psi
4,050,000 psi
Source: Ronald Shaeffer
University of Michigan, TCAUP
Structures II
Slide 9/23
Flexure and Shear in Beams
Reinforcement must be placed to resist
these tensile forces
In beams continuous over supports, the
stress reverses (negative moment).
In such areas, tensile steel is on top.
Shear reinforcement is provided by vertical
or sloping stirrups.
Cover protects the steel.
Adequate spacing allows consistent
casting.
University of Michigan, TCAUP
Structures II
Slide 10/23
Flexure – WSD Method
• Assumptions:
– Plane sections remain plane
– Hooke’s Law applies
– Concrete tensile strength is
neglected
– Concrete and steel are totally
bonded
• Allowable Stress Levels
– Concrete = 0.45f’c
– Steel = 20 ksi for gr. 40 or gr. 50
= 24 ksi for gr. 60
• Transformed Section
– Steel is converted to equivalent
concrete.
Es
n
Ec
Source: University of Michigan, Department of Architecture
University of Michigan, TCAUP
Structures II
Slide 11/23
Flexure Analysis
Procedure:
1. Assume the section is cracked to
the N.A
2. Determine the modular ratio:
Es
n
Ec
3. Transform the area of steel to
equivalent concrete, nAs
4. Calculate the location of the N.A.
using the balanced tension and
compression to solve for x.
Ac x c  At xt
5. Calculate the transformed Moment
of Inertia.
6. Calculate a maximum moment
based first on the allowable conc.
stress and again on the allowable
steel stress.
7. The lesser of the two moments will
control.
University of Michigan, TCAUP
b
fc
Ac
x
_
xc
N.A.
d
_
xt
d-x
nAs
fs/n
As
Ac x c  At x t
x
bx  nAs d  x 
2
b 2
x  nAs x  nAs d  0
2
bx 3
2
I tr 
 nAs d  x 
3
Structures II
Mc 
f c I tr
cc
cc  x
Ms 
f s I tr
nct
ct  d  x
Slide 12/23
Example – Flexure Analysis
1. Assume the section is
cracked to the N.A.
2. Determine the
transformation ratio, n
3. Transform the area of
steel to equivalent
concrete, nAs
Source: University of Michigan, Department of Architecture
University of Michigan, TCAUP
Structures II
Slide 13/23
Example – Flexure Analysis
cont.
4. Calculate the N.A. using
the balanced tension
and compression to
solve for x.
Acxc = Atxt
Source: University of Michigan, Department of Architecture
University of Michigan, TCAUP
Structures II
Slide 14/23
Example - Flexure Analysis
cont.
5. Calculate the
transformed Moment
of Inertia.
Source: University of Michigan, Department of Architecture
University of Michigan, TCAUP
Structures II
Slide 15/23
Example – Flexure Analysis
cont.
6. Calculate a
maximum moment
based first on the
allowable concrete
stress and again on
the allowable steel
stress.
7. The lesser of the
two moments will
control.
Source: University of Michigan, Department of Architecture
Source: University of Michigan, Department of Architecture
University of Michigan, TCAUP
Structures II
Slide 16/23
Effect of r
The behavior of the beam at failure (mode of failure)
is determined by the relative amount of steel present
– measured by r.
As
r
bd
r=0
No steel used. Brittle (sudden) failure.
r min
Just enough steel to prevent brittle failure
r min
200

fy
r < r balance
Steel fails first – ductile failure (desirable)
0.18 f c'
r 
fy
r balance = r max
r max  r balanced
Steel and concrete both stressed to allowable limit
r > r balance
Concrete fails first – brittle failure (not desirable)
University of Michigan, TCAUP
Structures II
Slide 17/23
Calculate r balance
Procedure:
1. Draw stress diagram using allowable
stresses fc and fs/n
2. Use similar triangles to find x and
bar_xs
3. Find bar_xc = x/2
4. Use moments of areas on transformed
section to solve for As.
5. Calculate r bal = As/bd
University of Michigan, TCAUP
Structures II
Slide 18/23
As
r
bd
“Internal Couple” Method
• Uses the internal force couple T & C to
determine the moment
• Defines factors k and j that can be
used to find depth of stress block and
moment arm of couple
• Provides equations for analysis or
design.
University of Michigan, TCAUP
Structures II
Es
n
Ec
Slide 19/23
Analysis by “Internal Couple”
Example :
1.
2.
3.
4.
5.
Find r =As/bd
Find k
Calculate j
Calculate either force T or C
Calculate M using either T or C
University of Michigan, TCAUP
Structures II
Slide 20/23
Flexure Design
Procedure:
1. Determine load conditions.
• choose material grade, f’c
• calculate n = Es/Ec
• estimate size, choose b and
estimate d
1 b 2
 
2 d 3
• determine loads (+ member DL)
• calculate moment
2. Choose a target steel ratio, ρ.
3. Sketch the stress diagram with
force couple.
4. Calculate d based on the required
moment.
5. Calculate As.
6. Choose bar sizes and spacing.
7. Choose beam size and revise
(back to step 1 with new b, d and ρ)
. University of Michigan, TCAUP
Structures II
Slide 21/23
Example – Flexure Design
Source: University of Michigan, Department of Architecture
1. As a simplification the
moment is given = 200 ft-k.
d will be determined based
on the moment.
f'c is given as 4000 psi
n is found = 8.
University of Michigan, TCAUP
Structures II
Slide 22/23
Example – Flexure Design
cont.
2. Steel ratio, As/bd is taken
as balanced for this
problem.
3. Using similar triangles,
determine depth of
reinforcement, D in
relationship to depth of
compression zone, x.
Calculate the compression
zone resultant, Rc in terms
of x
Rc = fcBx/2
4. Use the internal moment
couple
M = Rc(D-x/3)
to solve for x and D.
University of Michigan, TCAUP
Source: University of Michigan, Department of Architecture
Structures II
Slide 23/23
Example - Flexure Design
cont.
5. Calculate As using
Rc = Rt and
Rt = Asfs
6. Choose bar sizes and
spacing.
•
•
•
•
Area >= As
c.g. = D
must be symetric
minimum spacing
7. Choose cover,
recalculate dead load,
iterate with new
moment.
Source: ACI-318-05
University of Michigan, TCAUP
Structures II
Slide 24/23